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So when considering the situation of a rolling mirrored toiletpaper roll, because the camera is always going to see the reflection off a tangential plane that is co-moving with the axel, but not co-rotating with the wheel, we can model any infinitessimal part of the surface of the wheel that connects the raypath from the light source to the ray path from the camera as if it were a flat mirror moving with some velocity v, paralell to the ground, with some angle to the ground, and the velocity v is the velocity of the axel added to the component of the tangential velocity that is paralell to the ground.

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I believe looking at it in terms of an infinite number of mirrors is just fine. The problem is that if the mirror's orientation is anything but zero wrt its velocity there will be a detectable Doppler shift. The only place the orientation angle is zero is at the top and bottom of the wheel, which would show no Doppler.

I believe looking at it in terms of an infinite number of mirrors is just fine. The problem is that if the mirror's orientation is anything but zero wrt its velocity there will be a detectable Doppler shift. The only place the orientation angle is zero is at the top and bottom of the wheel, which would show no Doppler.

This is my position as well - this represents the 'special case' that I have readily admitted right from my first comment on the matter.

In the case of the mirror at the bottom of the cycle, it's because the mirror is stationary, in the case of the top of the cycle, it's because the red shift and the blue shift are equal and inverse, not because it is stationary.

Both of which display perfect specular reflection, and are moving between a light source and a camera, with some velocity \( v \leq \omega r\), while rotating about their C[sub]n[/sub] axis, with some angular velocity \(\omega\), oriented so that the plane containing their nC[sub]2[/sub] axes also contains the light source and the camera (or. alternatively, oriented so that the vector of motion is perpendicular to the C[sub]n[/sub] axis - same thing, different wording).

And now that the problem has been defined unambiguously, sensible discussion of the two seperate problems, or components of the problem, can begin.